What does it mean for two numbers to be coprime?

What does it mean for two numbers to be coprime?

In number theory, two integers $a$ and $b$ are said to be coprime (or relatively prime) if their only positive common divisor is $1$.

Mathematical Definition

Formally, $a$ and $b$ are coprime if:

$$\text{gcd}(a, b) = 1$$

For example, consider the numbers $8$ and $15$:

• The divisors of $8$ are $1, 2, 4,$ and $8$.
• The divisors of $15$ are $1, 3, 5,$ and $15$.

The only common divisor between $8$ and $15$ is $1$. Therefore, $8$ and $15$ are coprime.

Key Properties and Facts

1. Prime numbers are not required to be prime themselves: Neither $8$ nor $15$ are prime numbers, yet they are coprime to each other.
2. Consecutive integers: Any two consecutive integers (like $9$ and $10$, or $100$ and $101$) are always coprime.
3. Primes and composites: A prime number is coprime to any number it does not divide. For instance, $7$ is coprime to $10$, $11$, $12$, etc.
4. Fractions: Two numbers are coprime if and only if the fraction formed by them ($a/b$) is in its simplest form and cannot be reduced further.